Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [projective-geometry]

Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

12
votes
0answers
542 views

Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
7
votes
0answers
102 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
6
votes
0answers
95 views

Resolutions in Algebraic Geometry

I guess that the answer to this question can be given using Gröbner basis among many other computational methods, but my goal is to see if there is a more elementary way of approaching this problem. ...
6
votes
0answers
150 views

How can I compute the differential of a projective morphism?

I want to understand how to compute the differential of a projective morphism of schemes. For example, consider the projective morphism $$ \begin{matrix} & \textbf{Proj} \left(\frac{\mathbb{C}[s,t]...
6
votes
0answers
293 views

Bridging the gap between classical and modern projective geometry

The language of projective geometry is quite common in modern mathematics. For this reason I'd like to learn this subject, however, modern treatments are incredibly abstract. Now, I'm vaguely aware of ...
6
votes
0answers
254 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
5
votes
0answers
163 views

The Universal Property of Projective Space

Since the theories of affine and projective geometry are specified by certain axioms, we can consider the category $\mathcal{A}$ and $\mathcal{P}$ of affine and projective geometries, whose morphisms ...
5
votes
0answers
298 views

Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
5
votes
0answers
214 views

How to calculate the dimension of the intersection of projection of varieties?

In $\mathbb P_{n+1}$ we consider $d$ varieties $V_1,\ldots V_d$, each of them is defined by $d-1$ equations, as follows: we have $d-1$ polynomials $f_2(X_0,\ldots,X_n),\ldots,f_d(X_0,\ldots,X_n)$ and $...
4
votes
0answers
161 views

Fixed points of an involution

Let $V=\mathbb C^{2n}$ with the standard basis $\{e_1,e_2, \cdots , e_{2n}\}$ and let $\sigma$ be the involution $e_i \mapsto -e_{2n+1-i}$. This induces an involution of the Grassmannian $G(n,2n)$ of $...
4
votes
0answers
50 views

A question on the proof of Thm 1.4 in Algebraic Geometry A First Course

Thm 1.4 States that If $\Gamma\subset P^n$ is any collection of $d\leq 2n$ points in general position, then $\Gamma$ can be described by quadratic polynomials $\{f_i\}$. The proof intends to show ...
4
votes
0answers
140 views

A generalization of Zeeman-Gossard perspector theorem

I found a conjecture generalizing the Zeeman Gossard theorem a year ago, but I haven't found a solution for this conjecture. I'm an electrical engineer, I am not a mathematician. I don't know how to ...
4
votes
0answers
194 views

Invariant points and lines under homography

Given a matrix representation of an homography in a real projective space $P(\mathbb{R^3})$, what is the general procedure to calcule the invariant subspaces? A brief description would be enough.
4
votes
0answers
52 views

Morphism between surfaces

Suppose that $S$ is a surface of general type. Let $K_S$ the canonical bundle of $S$ and $\phi=\phi_{K_S}$ the canonical map. Suppose that the canonical map is a morphism from $S$ to $\mathbb{P}^{p_g-...
4
votes
0answers
187 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put $...
4
votes
0answers
193 views

Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with $F:\mathbb{...
4
votes
0answers
88 views

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
4
votes
0answers
609 views

Gluing construction of the projective space scheme.

When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, \tfrac{X_n}{X_i}\...
4
votes
0answers
141 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
votes
0answers
113 views

Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group $G_\...
4
votes
0answers
742 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
4
votes
0answers
122 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
4
votes
0answers
57 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
4
votes
0answers
179 views

Geometric interpretation of a certain property of graded rings

Let $k$ be a field, and $A$ a (commutative associative unital) $k$-algebra. It is known that there is a natural isomorphism between ‘regular’ (in the sense of regular maps of varieties) actions of $k^\...
4
votes
0answers
135 views

The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
3
votes
0answers
39 views

Finite projective plane order 11

Consider a finite projective plane of order 11, which is there more of: a) Unordered 4-tuples of lines with a non-empty intersection? b) Unordered 7-tuples of points which belong to the same line? ...
3
votes
0answers
96 views

Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
3
votes
0answers
48 views

Connection between number theory and projective geometry?

Consider the directed graphs $G_n^p$ with node set $\{0,1,\dots,p-1\}$, $0 \leq n < p$ with an arrow from $a$ to $b$ if $na=b\operatorname{mod}p$. These graphs are graphical multiplication tables ...
3
votes
0answers
102 views

Calculating singular points of a quintic curve in the projective plane?

I have the following question for an assignment. "An irreducible quintic curve in the real projective plane $P^2(R)$ is defined by $F: (X^2-Z^2)^2Y-(Y^2-Z^2)^2X=0$ Verify that the quartic curve ...
3
votes
0answers
68 views

Lines lying in a projective algebraic set

I want to find all lines lying entirely in the projective algebraic set $XY-ZW=0$ in $\mathbb{P}^3$, where $X, Y, Z, W$ are homogeneous coordinates. How can I do this?
3
votes
0answers
73 views

A question on Grassmannian

Let $T: Gr(n,\mathbb C^{2n}) \rightarrow G(n,\mathbb C^{2n})$ be the involution defined by $W \rightarrow W^{\perp}$ with respect to a symplectic form on $\mathbb C^{2n}$. Is there a direct proof (...
3
votes
0answers
355 views

Tough Olympiad geometry problem

$O$ is the point inside triangle $ABC$ . The lines joining the three vertices $A, B, C$ to $O$ cut the opposite sides in $K, L$, and $M$ respectively. A line through $M$ parallel to $KL$ cuts the ...
3
votes
0answers
153 views

Homogenous Coordinate Ring of Symmetric Product of Projective Space

I am trying to compute the homogenous coordinate ring of Sym$^2$P$^2$. I know that this is equal to $k[$P$^2 \times$P$^2]^{S_2}$ where $S_2$ acts on $k[$P$^2 \times$P$^2]$ by simultaneously switching $...
3
votes
0answers
154 views

Cross ratio: two different views

There are I think at least two views on the cross ratio of 4 points even https://en.wikipedia.org/wiki/Cross-ratio gives two definitions And I was wondering what is the relation between the two The ...
3
votes
0answers
36 views

Projective transformations of $\mathbb{P}^1$ and cross-ratio

Let $P=\{p_1,\ldots,p_6\}\subset\mathbb{P}^1$ be a set of six general points of the projective line. In particular there are no two different subsets $\{p_{i_1},\ldots,p_{i_4}\}$ and $\{p_{j_1},\ldots,...
3
votes
0answers
92 views

Natural line bundle over $\mathbb{P}^n$

What follows is from the book "Mirror Symmetry" by Hori et. al. From the definition of $\mathbb{P}^n$ we see there is a natural line bundle over $\mathbb{P}^n$ whose fiber over a point $l$ in $\...
3
votes
0answers
59 views

Conics meeting 8 general lines

I am trying to show that the number of plane conics in $\mathbb{P}^3$ meeting $8$ general lines is $92$, using what I know about intersection theory. I started considering the tautological bundle $S$ ...
3
votes
0answers
85 views

Genus of projective curves

I have the projective curve in $\mathbb{P}^2$ given by \begin{align} F(X,Y,Z)=Y^2 Z^2-X^4-Y^4. \end{align} I want to calculate the genus of the curve. My approach would be to calculate the partial ...
3
votes
0answers
457 views

Irreducible cubic curve with either a node or a cusp

I'm working on the following question from Shafarevich's Basic Algebraic Geometry I: Prove that an irreducible cubic curve $C$ has at most one singular point, and that the multiplicity of a ...
3
votes
0answers
119 views

Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate student need to know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm ...
3
votes
0answers
177 views

Question regarding Geometric meaning of Noether normalization theorem for projective varieties

In Ernst Kunz's ''commutative algebra and algebraic geometry'' book, ch.2, proposition 4.5 the author states: Let $K$ be an algebraically closed field, $V\subset \mathbb{P}^n(K)$ a variety of ...
3
votes
0answers
65 views

Tricks to find $\dim H^0(O(D)(n))$

Let $X$ be a curve of genus $g\neq 0$ and $D$ a divisor on $X$. If $n \in \mathbb{Z}$ and $O(D)(n)=O(D)\otimes O(n)$ is the twist of $O(D)$ by $n$, there is a manageable proceedings to find the ...
3
votes
0answers
84 views

Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
3
votes
0answers
381 views

Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
3
votes
0answers
274 views

line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it's right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in ...
3
votes
0answers
296 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let $...
3
votes
0answers
149 views

Projective Geometry Question.

I want to learn projective gemetry. I have the coxeter book and found some youtube videos. I have just started. I have tried answering this Challege question from the video, and so far. I have ...
3
votes
0answers
396 views

Finding the equations of a variety under a projection map

Suppose I have a projective variety $X$ in $\mathbb{P}^N$ ($N >2$, say) defined as the zero set of some homogeneous polynomials $f_1, \ldots, f_r$. Consider the projection map $[x_0: x_1 : \cdots :...
3
votes
0answers
531 views

Irreducible closed subsets of projective varieties

I want to prove the following lemma: Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety. My idea is as ...
3
votes
0answers
70 views

Do we have homogeneous coordinates for probabilities?

As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...